STUDY OF AXISYMMETRIC NATURE IN 3-D SWIRLING FLOW IN A CYLINDRICAL ANNULUS WITH A TOP ROTATING LID UNDER THE INFLUENCE OF AXIAL TEMPERATURE GRADIENT OR AXIAL MAGNETIC FIELD

Abstract

The three dimensional swirling flow has been obtained by solving Navier Stokes equations, expressed in cylindrical coordinate system, using finite difference technique on a staggered grid. An explicit finite difference method using pressure correction technique, for the solution of Navier-Stokes has been implemented to solve three dimensional flows.   Present study explores the 3-D axisymmetric nature of stratified swirling flow and vortex breakdown in a cylindrical annulus cavity with top rotating lid. The annulus is obtained by inserting a thin coaxial rod in cylindrical cavity. This rod may be stationary or rotating depending on the particular study. Three dimensional swirling flows in annuli have also been studied subjected to axial temperature gradient or under the influence of axial magnetic field. Influence of governing parameters Re, Ri and Ha on the overall heat transfer has been investigated through variation of the average Nusselt number with these parameters. Further, the present numerical results are shown to be in good agreement with the available benchmark solutions under the limiting conditions.

Keywords

• Three Dimensional,Swirling Flows,Magneto Hydrodynamics,Stratified Flow,Incompressible Fluid

References

1. [1] Escudier, M. P. "Observations of the flow produced in a cylindrical container by a rotating endwall." Experiments in fluids 2, no. 4 (1984): 189-196.
2. [2] Hourigan, K., L. J. W. Graham, and M. C. Thompson. "Spiral streaklines in pre‐vortex breakdown regions of axisymmetric swirling flows." Physics of Fluids (1994-present) 7, no. 12 (1995): 3126-3128.
3. [3] Stevens, José L., Z. Z. Celik, B. J. Cantwell, and J. M. Lopez. "Experimental study of vortex breakdown in a cylindrical, swirling flow." (1996).
4. [4] Fujimura, Kazuyuki, Hide S. Koyama, and Jae Min Hyun. "Time-dependent vortex breakdown in a cylinder with a rotating lid." Journal of fluids engineering 119, no. 2 (1997): 450-453.
5. [5] Spohn, A., M. Mory, and E. J. Hopfinger."Experiments on vortex breakdown in a confined flow generated by a rotating disc." Journal of Fluid Mechanics370 (1998): 73-99.
6. [6] Blackburn, Hugh M., and J. M. Lopez. "Symmetry breaking of the flow in a cylinder driven by a rotating end wall." Physics of Fluids 12, no. 11 (2000): 2698-2701.
7. [7] Sotiropoulos, Fotis, and Yiannis Ventikos. "The three-dimensional structure of confined swirling flows with vortex breakdown." Journal of Fluid Mechanics 426 (2001): 155-175.
8. [8] Vogel, H. U. 1968: Experimentelle Ergebnisse t~ber die laminare Str6mung in einem zylindrischen Geh~iuse mit darin rotierender Scheibe. MPI Bericht 6.

9. [9] Ronnenberg, B. 1977: Ein selbstjustierendes 3-Komponenten-Laserdoppleranemometer nach dem Vergleichsstrahlverfahren,angewandt far Untersuchungen in einer station~ tren zylinder symmetrischen Drehstr6mung mit einem Rt~ckstromgebiet. MPI Bericht 20.
10. [10] Escudier, M. P., and J. J. Keller. Vortex breakdown: a two-stage transition.BROWN BOVERI RESEARCH CENTER BADEN (SWITZERLAND), 1983.
11. [11] Lopez, J. M. (1990). Axisymmetric vortex breakdown Part 1. Confined swirling flow. Journal of Fluid Mechanics, 221, 533-552.
12. [12] Lugt, H.J. and Abboud, M., 1987, “Axisymmetric vortex breakdown with and without temperature effects in a container with a rotating lid,” Journal of Fluid Mechanics., 179, pp.179-200.
13. [13] Bessaïh R, Marty P, Kadja M. Numerical study of disk driven rotating MHD flowof a liquid metal in a cylindrical enclosure. Acta Mech 1999; 135:153.
14. [14] Bessaïh R, Kadja M, Eckert K, Marty P. Numerical and analytical study of rotating flow in an enclosed cylinder under an axial magnetic field. Acta Mech2003; 164:175.
15. [15] Gelfgat YM, Gelfgat AY. Experimental and numerical study of rotating magnetic field driven flow in cylindrical enclosures with different aspect ratios. Magnetohydrodynamics.2004; 40:147.
16. [16] Lee, C. H., & Hyun, J. M. (1999). Flow of a stratified fluid in a cylinder with a rotating lid. International journal of heat and fluid flow, 20(1), 26-33.
17. [17] Kim, W. N., & Hyun, J. M. (1997). Convective heat transfer in a cylinder with a rotating lid under stable stratification. International Journal of Heat and Fluid Flow, 18(4), 384-388.
18. [18] Iwatsu, R. (2004). Flow pattern and heat transfer of swirling flows in cylindrical container with rotating top and stable temperature gradient. International journal of heat and mass transfer, 47(12), 2755-2767.
19. [19] Chen, S. (2011). Entropy generation inside disk driven rotating convectional flow. International Journal of Thermal Sciences, 50(4), 626-638.
20. [20] Dash S., and Singh N., 2016, “Effects of Partial Heating of Top Rotating Lid With Axial Temperature Gradient On Vortex Breakdown In Case Of Axisymmetric Stratified Lid Driven Swirling Flow,” Yildiz Technical University Press, Istanbul, Turkey , J. Thermal Eng., 2(Sp. Issue 4), pp. 883-896.
21. [21] Gefagat A. Y. , Destabilization of free convection by weak rotation, 9th international conference heat transfer fluid mechanics and thermodynamics,16-18 july 2012,Malta.
22. [22] Bessaïh R, Boukhari A, Marty P. Magnetohydrodynamics stability of a rotatingflow with heat transfer. Int Commun Heat Mass 2009; 36:893.
23. [23] Mahfoud B, Bessaih R. Oscillatory swirling flows in a cylindrical enclosure with co-/counter-rotating end disks submitted to a vertical temperature gradient. Fluid Dynamics & Materials Processing. 2012; 8:1.
24. [24] Verzicco, R., and P. Orlandi. "A finitedifference scheme for three-dimensional incompressible flows in cylindrical coordinates." Journal of Computational Physics 123, no. 2 (1996): 402-414.
25. [25] Barbosa, Emerson, and Olivier Daube. "A finite difference method for 3D incompressible flows in cylindrical coordinates." Computers & fluids 34, no. 8 (2005): 950-971.
26. [26] Fukagata, Koji, and Nobuhide Kasagi. "Highly energy-conservative finite difference method for the cylindrical coordinate system." Journal of Computational Physics 181, no. 2 (2002): 478-498.
27. [27] Chorin, A. J., A Numerical method for solving incompressible viscous flow problems, Journal of Comput. Phys., Vol 2, pp 12-26, 1967.
28. [28] Peyret, R.&Taylor, D.(1983). Computational methods for fluid flow;SprigerVerlag.